Let `f(x+y)=f(x)+f(y)AA x, y in R` If `f(x)` is continous at x = 0, then `f(x)` is continuous at

Options

A. all natural numbers only
B. all integers only
C. all rational numbers only
D. all real numbers.

Answer

To solve the problem, we need to show that the function \( f(x) \) is continuous at every point in \( \mathbb{R} \) given that it satisfies the functional equation \( f(x+y) = f(x) + f(y) \) for all \( x, y \in \mathbb{R} \) and is continuous at \( x = 0 \).

### Step-by-Step Solution:

1. **Understanding the Functional Equation**:
The equation \( f(x+y) = f(x) + f(y) \) suggests that \( f \) is a linear function. This is a well-known property of functions that satisfy this equation.

2. **Finding \( f(0) \)**:
Set \( x = 0 \) and \( y = 0 \) in the functional equation:
\[ f(0 + 0) = f(0) + f(0) \implies f(0) = 2f(0). \]
This implies \( f(0) = 0 \).

3. **Using Continuity at \( x = 0 \)**:
Since \( f(x) \) is continuous at \( x = 0 \), we have:
\[ \lim_{x \to 0} f(x) = f(0) = 0. \]

4. **Finding Continuity at an Arbitrary Point \( k \)**:
Let \( k \) be any point in \( \mathbb{R} \). We want to show that \( f(x) \) is continuous at \( x = k \). We will check the left-hand limit and right-hand limit at \( k \).

5. **Left-Hand Limit**:
The left-hand limit as \( x \) approaches \( k \) is:
\[ \lim_{x \to k^-} f(x) = \lim_{h \to 0} f(k - h) = \lim_{h \to 0} (f(k) + f(-h)) = f(k) + \lim_{h \to 0} f(-h). \]
Since \( f(-h) = -f(h) \) and \( \lim_{h \to 0} f(h) = 0 \), we have:
\[ \lim_{h \to 0} f(-h) = 0 \implies \lim_{x \to k^-} f(x) = f(k). \]

6. **Right-Hand Limit**:
The right-hand limit as \( x \) approaches \( k \) is:
\[ \lim_{x \to k^+} f(x) = \lim_{h \to 0} f(k + h) = \lim_{h \to 0} (f(k) + f(h)) = f(k) + \lim_{h \to 0} f(h). \]
Again, since \( \lim_{h \to 0} f(h) = 0 \):
\[ \lim_{x \to k^+} f(x) = f(k). \]

7. **Conclusion**:
Since both the left-hand limit and right-hand limit at \( k \) equal \( f(k) \), we conclude that:
\[ \lim_{x \to k} f(x) = f(k). \]
Therefore, \( f(x) \) is continuous at \( x = k \).

8. **Generalization**:
Since \( k \) was an arbitrary point in \( \mathbb{R} \), we conclude that \( f(x) \) is continuous for all \( x \in \mathbb{R} \).

### Final Answer:
Thus, \( f(x) \) is continuous at all real numbers.

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